In the prior art, nonminimum-phase systems could only be controlled by using low bandwidth, sluggish controllers. In layman's terms, nonminimum-phase systems move in a direction initially opposite to the direction they are pushed. Such systems arise when, for example, one attempts to dampen vibrations on a flexible structure in optical instruments by acting at a location remote from the flexible structure in the case of a "noncolocated" control system.
Nonminimum-phase systems arise in many aerospace applications of noncolocated control, such as spacecraft slewing, instrument pointing/tracking, flexible robotics, acoustic systems, chemical process control systems, compact disk controllers, floppy disk controllers and even real-time equalization of nonminimum-phase communication systems. An object of this invention is to achieve high performance control for the class of nonminimum-phase systems that encompasses these and other applications which are referred to generically as "plants."
A Zero Annihilation Periodic (ZAP) control law was introduced by the present inventor, David S. Bayard, in publications titled "Globally stable adaptive periodic control," Jet Propulsion Laboratory, Internal Document JPL D-9448, February 3, 1992 and "Zero annihilation methods for direct adaptive control of nonminimum-phase systems," Proc. Seventh Yale Workshop on Adaptive and Learning Systems, Yale University, May 1992, for controlling nonminimum-phase systems using stable inversion of the plant. The general ZAP approach is based on the notion of a mathematical "lifting" performed on the plant input and output signals with mapping between the vectorized input and output quantities.
A key property of the ZAP liftings which makes them so useful is that the transmission zeros of the lifted and vectorized plant input signal are annihilated (i.e., placed to the origin). This zero-annihilation (ZA) property allows the vectorized plant signals to be stably inverted using standard control methods. The result is important to many areas of control, communications, and signal processing where a stable plant inverse is often desired but not possible due to nonminimum-phase restrictions.
In a paper by R. Lozano-Leal, "Robust adaptive regulation without persistent excitation," IEEE Trans. Automatic Control, Vol. 34, pp. 1260-1267, December 1989, a multirate sampling method is presented which allows stable inversion of any linear time-invariant finite-order plant signal. Not surprisingly Lozano's lifting has also been applied to developing stable adaptive control algorithms for nonminimum phase systems. See also R. Lozano et al., "Singularly-free adaptive pole placement using periodic controllers," IEEE Trans. Automatic Control, Vol. 38, pp. 104-108, January 1993.
Lifting in this invention is an extension of the prior art represented by Bayard, supra, and Lozano-Leal, supra, to "extended horizons," i.e., to extended periodic windows over which liftings (samples) are taken in adaptive controllers of the type described by Lozano et al., supra, for example. Extended horizon liftings are crucial for control gain reduction in order to allow practical implementations of control in nonminimum-phase systems. The distinction between extended horizon liftings of the present invention and the prior art will now be described.
Since Lozano-Leal's liftings are distinct from the present invention in that it utilizes a horizon size of N=2n, where n is the plant order and N is the length of the window for the liftings, and the number of elements in the system input equals those in the system output (i.e., .sigma..sub.u =.sigma..sub.y), it will be denoted as the "2n-lifting," while in the present invention N&gt;n and preferably N.gtoreq.2n-1 in a large class of systems in which .sigma..sub.u &gt;.sigma..sub.y for output tracking (mapping), such as in feedback control systems, and .sigma..sub.u &lt;.sigma..sub.y for input tracking, such as in communications systems, and which enjoy the same zero annihilation properties as N=2n. Unlike the special case of Lozano-Leal's and other prior art liftings, the present invention allows the use of all extended horizon liftings with N greater than n and preferably equal to or greater than 2n-1 with .sigma..sub.u &gt;.sigma..sub.y or output tracking (OT) and .sigma..sub.u &lt;.sigma..sub.y for input tracking (IT). An important consequence is that such extended horizon liftings lead to plant-inverse controllers with significantly reduced control gains. This overcomes a problem associated with Lozano-Leal's lifting, where N=2n and .sigma..sub.u =.sigma..sub.y, which has prevented its use in many applications of practical interest that the present invention will reach. To illustrate the present invention, a simulation example is provided below in which the peak control requirement is reduced by four orders of magnitude using extended horizon liftings.
It will also be shown that as a dual result, a related class of extended horizon liftings enables equalization of nonminimum-phase channels in communication systems. This overcomes the standard problem of inverting the channel in a stable fashion. In this invention, the extended horizon property allows channel inversion by least squares estimation, which provides smoothing in the case of noise.
As background information, consider a plant input/output model, ##EQU1## where the polynomials A and B are assumed to be relatively prime. It is assumed that b.sub.1 .noteq.0, so that the polynomial B can be factored uniquely into the form B(z.sup.-1)=z.sup.-d b.sub.1 B(z.sup.-1) where B(z.sup.-1) is a monic polynomial and d=1 is the plant delay. The choice d=1 is for convenience only and is not a fundamental restriction. In the case that d.noteq.1, all subsequent expressions can be appropriately modified without loss of generality.
Choose some horizon time N&gt;n. The system of Eq. (1) is iterated to give the following system of linear equations, EQU Y(k)=A.sub.1 Y(k)+A.sub.2 Y(k-1)+B.sub.1 U(k)+B.sub.2 U(k-1)(2)
where, ##EQU2## Y(k)=plant output u(k)=plant input
A.sub.1 =lower triangular Toeplitz, with first column [O,-a.sub.1, . . . ,-a.sub.n,O, . . . ,O] PA0 A.sub.2 =upper triangular Toeplitz, with first row [O,-a.sub.1, . . . ,-a.sub.n,O, . . . ,O] PA0 B.sub.1 =lower triangular Toeplitz, with first column [b.sub.1,b.sub.2, . . . ,b.sub.n,O, . . .O].sup.T PA0 B.sub.2 =upper triangular Toeplitz, with first row [O, . . . , O, b.sub.n, . . . , b.sub.2 ]
Example 1 Let n=3 and N=4. Then Eq. (2) becomes, ##EQU3##
It is convenient to combine terms involving Y(k) in eq. (2) and rearrange to give the following lifting of P. Albertos, "Block multirate input-output model for sampled-data control systems," IEEE Trans. Automatic Control, Vol. 35, No. 9, pp. 1085-1088, September 1990.
Albertos' Lifting: EQU Y(k)=AY(k-1)+HU(k)+BU(k-1) (4)
where, EQU A=(I-A.sub.1).sup.-1 A.sub.2 ( 5a) EQU H=(I-A.sub.1).sup.-1 B.sub.1 ( 5b) EQU B=(I-A.sub.1).sup.-1 B.sub.2 ( 5c)
It is noted that since A.sub.1 is lower triangular with zeros on the diagonal, the quantity (I-A.sub.1) is always invertible. Hence the quantities in Eq. (5) always exist.
Polynomial A is divided into B to give impulse response sequence {h.sub.i }, ##EQU4## The Markov parameter sequence {h.sub.i } is not assumed to be convergent (i.e., the system may be unstable). Using the Toeplitz structure of A.sub.1 and B.sub.1 and relation, (Eq. (6)), it can be shown that the matrix H in Eqs. (4) and (5) can be written in terms of the impulse response parameters, ##EQU5## This is the desired expression for H, i.e., H=lower triangular Toeplitz, with first column [h.sub.1,h.sub.2, . . . , h.sub.H ].sup.T. Since the delay is unity by assumption (i.e., d=1), the matrix H has a nonzero diagonal (i.e., h.sub.1 .noteq.0), and is always invertible.